# WHAT IS THE GRAPH OF $\,y = f(x)\,$?

• PRACTICE (online exercises and printable worksheets)

The web exercise Graphs of Functions in the Algebra II curriculum gives a thorough introduction to graphs of functions.
For those who need only a quick review, the key concepts are repeated here.
The exercises on this current web page duplicate those in Graphs of Functions.

• The equation ‘$\,y = f(x)\,$’ is an equation in two variables, $\,x\,$ and $\,y\,$.
The graph of the equation $\,y = f(x)\,$ is the picture of all the points $\,(x,y)\,$ that make it true;
observe that to make this equation true, $\,y\,$ must equal $\,f(x)\,$.
Thus, the graph of the equation $\,y = f(x)\,$ is the set of all points of the form $\,\color{green}{\bigl(x,\overbrace{f(x)}^{y}\bigr)}\,$.
• The graph of a function $\,f\,$ is the set of all its $\,\bigl(\text{input},\text{output}\bigr)\,$ pairs;
recall that when $\,x\,$ is an input, then $\,f(x)\,$ is the corresponding output.

Thus, the graph of $\,f\,$ is the set of all points of the form $\,\color{green}{\bigl(x,f(x)\bigr)}\,$.
• Compare!
The following two requests ask for exactly the same thing:
-- Graph the equation $\,\color{green}{y = f(x)}\,$.
-- Graph the function $\,\color{green}{f}\,$.
Both are asking for the set of all points of the form $\,\bigl(x,f(x)\bigr)\,$.

## Understanding the Relationship between an Equation and its Graph

There are things that you can DO to an equation $\,y = f(x)\,$ that will change its graph.
Or, there are things that you can DO to a graph that will change its equation.
Stretching, shrinking, moving up/down/left/right, reflecting about axes;
they're all covered thoroughly in the next few web exercises:

An understanding of these graphical transformations makes it easy to graph a wide variety of functions,
by starting with a basic model and then applying a sequence of transformations to change it to the desired function.

For example, after mastering the graphical transformations, you'll be able to do the following:

• Need the graph of $\,y = -\sqrt{3x} + 2\,$? No problem!
Start with the graph of $\,y = \sqrt{x}\,$ (one of the basic models):
A) divide all the $\,x\,$-values by $\,3\,$ (a horizontal compression/shrink—the points get closer to the $y$-axis);
B) then reflect about the $x$-axis;
C) then move it up $2$
You'll understand why the order BAC would also work, but the order CAB doesn't!
• Or, suppose you have the graph (call it $\,G\,$) of a known equation $\,y = f(x)\,$.
The graph is being changed, though, and you need the corresponding new equations.
-- If $\,G\,$ is shifted up $\,2\,$ units, the new equation is $\,y = f(x) + 2\,$. (Transformations involving $\,y\,$ are intuitive.)
-- If $\,G\,$ is shifted right $\,2\,$ units, the new equation is $\,y = f(x-2)\,$. (Transformations involving $\,x\,$ are counter-intuitive.)
-- If $\,G\,$ is reflected about the $y$-axis, the new equation is $\,y = f(-x)\,$.

For your convenience, all the graphical transformations are summarized in the GRAPHICAL TRANSFORMATIONS table below.
Given any entry in a row, you should (eventually!) be able to fill in all the remaining entries in that row.

# SUMMARY: GRAPHICAL TRANSFORMATIONS

SET-UP FOR THE TABLE:
• you're starting with the equation $y = f(x)$
(so, the ‘previous $\color{purple}{y}$-value’—see the first column below—is $\,f(x)$)
• assume:
$p > 0$     (‘$p$’ for Positive)
$g > 1$     (‘$g$’ for Greater than)
• the point $\,(a,b)\,$ is a point on the graph of $\,y = f(x)\,$, so that the equation $\,b = f(a)\,$ is true
 TRANSFORMATIONS INVOLVING $\,\boldsymbol{y}$ (note that transformations involving $\,y\,$ are intuitive) DO THIS TO THE PREVIOUS $y$-VALUE NEW EQUATION NEW GRAPH $(a,b)$ MOVES TO ... TRANSFORMATION TYPE add $\,p$subtract $\,p$ $y = f(x) + p$$y = f(x) - p shifts \,p\, units UPshifts \,p\, units DOWN (a,b+p)$$(a,b-p)$ vertical translationvertical translation multiply by $\,-1$ $y = -f(x)$ reflect about $x$-axis $(a,-b)$ reflection about $x$-axis multiply by $\,g$ $y = g\cdot f(x)$ vertical stretchby a factor of $\,g$ $(a,gb)$ vertical stretch/elongation divide by $\,g$ $\displaystyle y = \frac{f(x)}{g}$ vertical shrinkby a factor of $\,g$ $\displaystyle \bigl(a,\frac{b}{g}\bigr)$ vertical shrink/compression take absolute value $y = |f(x)|$ part below $x$-axis flips up $(a,|b|)$ absolute value

 TRANSFORMATIONS INVOLVING $\,\boldsymbol{x}$ (note that transformations involving $\,x\,$ are counter-intuitive) REPLACE ... NEW EQUATION NEW GRAPH $(a,b)$ MOVES TO ... TRANSFORMATION TYPE every $\,x\,$ by $\,x+p\,$every $\,x\,$ by $\,x-p$ $y = f(x+p)$$y = f(x-p) shifts \,p\, units LEFTshifts \,p\, units RIGHT (a-p,b)$$(a+p,b)$ horizontal translationhorizontal translation every $\,x\,$ by $\,-x\,$ $y = f(-x)$ reflect about $y$-axis $(-a,b)$ reflection about $y$-axis every $\,x\,$ by $\,gx\,$ $y = f(gx)$ horizontal shrinkby a factor of $\,g$ $\displaystyle \bigl(\frac{a}{g},b\bigr)$ horizontal shrink/compression every $\,x\,$ by $\,\displaystyle \frac{x}{g}\,$ $\displaystyle y = f\bigl(\frac{x}{g}\bigr)$ horizontal stretchby a factor of $\,g$ $(ga,b)$ horizontal stretch/elongation
Master the ideas from this section